Convergence of the Riemannian Langevin Algorithm

TL;DR

This paper extends the Langevin sampling algorithm to Riemannian manifolds, proving rapid convergence under certain conditions, and offers a new approach for sampling complex densities by leveraging geometric properties.

Contribution

It introduces a Riemannian version of the Langevin Algorithm with convergence guarantees on Hessian manifolds, enabling sampling of non-smooth densities via manifold geometry.

Findings

Proves rapid convergence of the Riemannian Langevin Algorithm on Hessian manifolds.

Reduces sampling of constrained densities in Euclidean space to smooth densities on manifolds.

Provides tools for sampling isoperimetric densities on polytopes using logarithmic barrier metrics.

Abstract

We study the Riemannian Langevin Algorithm for the problem of sampling from a distribution with density $\nu$ with respect to the natural measure on a manifold with metric $g$. We assume that the target density satisfies a log-Sobolev inequality with respect to the metric and prove that the manifold generalization of the Unadjusted Langevin Algorithm converges rapidly to $\nu$ for Hessian manifolds. This allows us to reduce the problem of sampling non-smooth (constrained) densities in ${\bf R}^n$ to sampling smooth densities over appropriate manifolds, while needing access only to the gradient of the log-density, and this, in turn, to sampling from the natural Brownian motion on the manifold. Our main analytic tools are (1) an extension of self-concordance to manifolds, and (2) a stochastic approach to bounding smoothness on manifolds. A special case of our approach is sampling isoperimetric densities restricted to polytopes by using the metric defined by the logarithmic barrier.